Question

here is a data set. 9,2,9,6,4 progress: 0/3 part 1 of 2 (a) find the mean, median, mode, range, and standard deviation of the data set above. round your answer for the standard deviation to the nearest hundredth. mean: 3 median: 6 mode: range: standard deviation:

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 5$, $x_1=9,x_2 = 2,x_3=9,x_4=6,x_5 = 4$. So, $\bar{x}=\frac{9 + 2+9+6+4}{5}=\frac{30}{5}=6$.

Step2: Calculate the median

First, arrange the data - set in ascending order: $2,4,6,9,9$. Since $n = 5$ (odd), the median is the middle value. So the median is $6$.

Step3: Calculate the mode

The mode is the value that appears most frequently in the data - set. Here, $9$ appears twice and other values appear once, so the mode is $9$.

Step4: Calculate the range

The range is the difference between the maximum and minimum values in the data - set. The maximum value is $9$ and the minimum value is $2$, so the range is $9 - 2=7$.

Step5: Calculate the standard deviation

The formula for the sample standard deviation $s$ is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$.
$(x_1-\bar{x})^2=(9 - 6)^2=9$, $(x_2-\bar{x})^2=(2 - 6)^2 = 16$, $(x_3-\bar{x})^2=(9 - 6)^2=9$, $(x_4-\bar{x})^2=(6 - 6)^2=0$, $(x_5-\bar{x})^2=(4 - 6)^2 = 4$.
$\sum_{i = 1}^{n}(x_i-\bar{x})^2=9 + 16+9+0+4=38$.
$s=\sqrt{\frac{38}{4}}=\sqrt{9.5}\approx3.08$.

Answer:

Mean: $6$
Median: $6$
Mode: $9$
Range: $7$
Standard deviation: $3.08$